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DanielLC comments on Game Theory As A Dark Art - Less Wrong
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I don't understand the dollar auction.
Alice and Bob both have the strategy "bid until you run out of money". Alice has $50, and the auction is currently at $20. If Alice and Bob continue their strategies, Alice has a 50% chance of losing the auction and losing $50, and a 50% chance of winning the auction and losing a smaller amount of money. Her expected loss is more than $25. If she changes her strategy to "never bid", she'll instead lose $20, which clearly isn't as bad. As such, this strategy is unstable, and not a Nash equilibrium.
The only Nash equilibria I can find in it are Alice always bids and Bob never bids, Bob always bids and Alice never bids, and a mixed strategy in which they may-or-may-not bid in any given round.
It's a tax on only thinking ahead 1 step.
You don't know when the other person will give up and it's better to be the last person to give up.
No, it's best to never get in to it in the first place - once you bid even $1, you can come out worse than you started (losing the bid), but if you never bid, you just break even.
If never giving up really is the best strategy, then you can assume they'll use it. If they're using it, it's better to give up now. If you assume they use a mixed strategy in which they have a chance of giving up, it really doesn't matter what you do, as is often the case with Nash equilibria, but you have to use the same mixed strategy or they'd change theirs.